Accessibility and occlusion of biopolymers, ray tracing of radiating tubes, and the temperature of a tangle

We introduce a measure of complexity, an energy, for any conformation of filaments. It is the occlusion, the portion hidden when viewed from an arbitrary exterior point. By inverting we get the exposure, a first approximation of the accessibility of the filaments. Assuming the filament is a source, we get the self-irradiation, which leads to both an interpretation as the temperature and a visualization technique: ray tracing as a virtual laboratory. There is a wide variety of applications, from enzyme action on and radiation damage of biopolymers, to the geometry of light bulb filaments. Energy minimization provides automatic detangling, resulting in symmetric and pleasing conformations.

Figure 1

(Color online) Tube geometry. Let the upper cylinder (red exterior in color image) be the tube element $C\left(x\right)$, the lower (blue) $C\left(y\right)$. The thin connecting cylinder between $C\left(x\right)$ and $C\left(y\right)$ is the separation vector $r$. [In the color image $C\left(x\right)$ is a source of blue light, which is why $C\left(y\right)$ is blue, as is the interior of $C\left(x\right)$. The exterior of $C\left(x\right)$ is red because it is reflecting the red light emanating from $C\left(y\right)$].

Figure 2

Occlusion and accessibility as the “inverse” of self-radiation and temperature. $2A$ (self-radiation) was created via ray-tracing. $2B$ is simply the inverse image via Photoshop, and accessible regions are light colored, occluded ones are “shadowed.” This illustration was suggested by Felice Frankel.

Figure 3

(Color online) Simplification of complex conformations to canonical forms by the gradient flow of $E$.

Figure 4

(Color online) Apparent energy minimums for some simple knot and link types.

Figure 5

Symmetry of figure 8 canonical conformation (energy minimum). Symmetry of $8_3$ energy minimum. These are rigidly achiral knots—they are topologically equivalent to their mirror images, and this is shown by rigid rotations of the canonical conformations.

Figure 6

(Color online) Ray-traced radiating tube pictures, including a coil conformation like a light bulb filament. In the multicolored images, each component (closed loop) radiates a different color light.